2013 UNCO Math Contest II Problems/Problem 11

Problem

(a) Stages $1$ and $2$ each contain $1$ tile. Stage $6$ contains $8$ tiles. If the pattern is continued, how many tiles will Stage $15$ contain?

(b) What is the first Stage in which the number of tiles is a multiple of $2013$ ?

$[asy] size(14cm,0); draw((0,0)--(1,1)--(2,1)--cycle,black); draw((3,0)--(4,2)--(5,0)--cycle,black); draw((6,0)--(7,2)--(8,0)--cycle,black); draw((6,0)--(5.5,-1)--(8,0)--cycle,black); draw((9,0)--(10,2)--(11,0)--cycle,black); draw((9,0)--(8.5,-1)--(11,0)--cycle,black); draw((8.5,-1)--(11,0)--(11.5,-1)--cycle,black); draw((13,0)--(14,2)--(15,0)--cycle,black); draw((13,0)--(12.5,-1)--(15,0)--cycle,black); draw((12.5,-1)--(15,0)--(15.5,-1)--cycle,black); draw((12.5,-1)--(11.5,-3)--(15.5,-1)--cycle,black); draw((12.5,-1)--(14,-1.75),black); draw((18,0)--(19,2)--(20,0)--cycle,black); draw((18,0)--(17.5,-1)--(20,0)--cycle,black); draw((17.5,-1)--(20,0)--(20.5,-1)--cycle,black); draw((17.5,-1)--(16.5,-3)--(20.5,-1)--cycle,black); draw((17.5,-1)--(21.5,-3)--(20.5,-1)--cycle,black); draw((16.5,-3)--(21.5,-3),black); draw((19,-1.75)--(20,-3),black); MP("Stage 1",(.5,-3)); MP("Stage 2",(3.5,-3)); MP("Stage 3",(6.5,-3)); MP("Stage 4",(10,-3)); MP("Stage 5",(14,-3)); MP("Stage 6",(19,-3)); [/asy]$

Solution

(a) $610$ (b) $60^{th}$

2013 UNCO Math Contest II (Problems • Answer Key • Resources)
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2013 UNCO Math Contest II Problems/Problem 11

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